Growth of Instabilities on Displacement Fronts in Porous Media

Abstract

The problem investigated is that of the penetration of a fluid into a porous medium containing a more viscous liquid. In order to do this, the flow potentials for a displacement front which is just about to become unstable are calculated. For such a displacement front it is possible to linearize the differential equations, and to give a description in terms of Fourier analysis. The law of growth for each spectral component of the front is deduced, and it is shown how the time dependence of the whole front can be represented by a superposition of elemental solutions. Subsequently, the effect of the heterogeneities contained in the porous medium is accounted for by introducing a random velocity perturbation term into the differential equation for each spectral component. In this fashion one arrives at an equation describing the growth of each spectral component of the fingers with time. It is shown that, under given external conditions, fingering should be independent of the speed with which the displacement proceeds. This is, in fact, what has been observed experimentally.